Euler-Poincare Equations: Fluids, Waves, and Vision

The Euler-Poincare equations were born in 1901 when Poincare made a sweeping generalization of the classical Euler equations for the rigid body and ideal fluids. He did this by formulating the equations on a general Lie algebra, with the rigid body being associated with the rotation Lie algebra and fluids with the Lie algebra of divergence free vector fields. Since then, this setting has been used for many other situations, such as the KdV equation, shallow water waves, averaged fluid equations, and the template matching equations of computer vision to name just a few. This talk will give an overview of this general approach and then will focus on the specifics for the case of the algebra of all vector fields. Special singular solutions will be described which generalize the peakon (soliton) solutions of the shallow water equations from one to higher dimensions; it will be shown that momentum maps (in the sense of Noether's theorem from mechanics) play an important role in these singular solutions. (Joint work with Darryl Holm)